# Properties

 Label 136.2.h.a Level $136$ Weight $2$ Character orbit 136.h Analytic conductor $1.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 136.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.08596546749$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 8 x^{14} + 20 x^{12} + 36 x^{10} + 240 x^{8} - 156 x^{6} + 268 x^{4} + 136 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + \beta_{12} q^{3} -\beta_{3} q^{4} -\beta_{6} q^{5} -\beta_{10} q^{6} + \beta_{9} q^{7} + \beta_{8} q^{8} + ( 1 - \beta_{2} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + \beta_{12} q^{3} -\beta_{3} q^{4} -\beta_{6} q^{5} -\beta_{10} q^{6} + \beta_{9} q^{7} + \beta_{8} q^{8} + ( 1 - \beta_{2} + \beta_{8} ) q^{9} + \beta_{1} q^{10} -\beta_{7} q^{11} + ( -\beta_{9} - \beta_{12} + \beta_{14} ) q^{12} + ( -\beta_{8} - \beta_{15} ) q^{13} + ( -\beta_{6} - \beta_{13} ) q^{14} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{15} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{15} ) q^{16} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{14} ) q^{17} + ( \beta_{11} + \beta_{15} ) q^{18} + ( -1 + \beta_{2} + \beta_{4} - \beta_{11} - \beta_{15} ) q^{19} + ( \beta_{1} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{20} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{11} + \beta_{15} ) q^{21} + ( -\beta_{1} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{22} + ( -\beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{23} + ( -\beta_{1} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{24} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} - \beta_{11} ) q^{25} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} + \beta_{11} ) q^{26} + ( 2 \beta_{6} + \beta_{10} + \beta_{13} - \beta_{14} ) q^{27} + ( \beta_{1} + \beta_{6} + \beta_{7} + \beta_{12} + \beta_{14} ) q^{28} + ( -\beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{29} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{15} ) q^{30} + ( -2 \beta_{1} + \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{31} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{8} - \beta_{15} ) q^{32} + ( -\beta_{3} + \beta_{5} + \beta_{11} ) q^{33} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{34} + ( -\beta_{3} + \beta_{5} + \beta_{8} - \beta_{11} + \beta_{15} ) q^{35} + ( -3 + 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} - 2 \beta_{11} - \beta_{15} ) q^{36} + ( \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{37} + ( 3 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{15} ) q^{38} + ( 2 \beta_{1} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{39} + ( -\beta_{1} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{40} + ( 2 \beta_{1} - \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{41} + ( -2 + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{11} - \beta_{15} ) q^{42} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{43} + ( -2 \beta_{6} + \beta_{9} + \beta_{12} + \beta_{14} ) q^{44} + ( \beta_{6} - 2 \beta_{12} ) q^{45} + ( -\beta_{1} - \beta_{7} - 2 \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{46} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{47} + ( -\beta_{1} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{48} + ( -1 + \beta_{3} - \beta_{5} - \beta_{11} ) q^{49} + ( 3 - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{15} ) q^{50} + ( \beta_{3} - \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{51} + ( -1 - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{15} ) q^{52} + ( \beta_{2} + \beta_{3} - 3 \beta_{5} + \beta_{8} + \beta_{11} ) q^{53} + ( -\beta_{1} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{14} ) q^{54} + ( 1 - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{11} ) q^{55} + ( -2 \beta_{7} - 2 \beta_{10} + 2 \beta_{12} ) q^{56} + ( \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{57} + ( \beta_{1} - 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{14} ) q^{58} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{8} ) q^{59} + ( -2 \beta_{2} + 2 \beta_{5} + 2 \beta_{11} ) q^{60} + ( -\beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{61} + ( -\beta_{1} - 2 \beta_{6} + \beta_{7} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{62} + ( -\beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{14} ) q^{63} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{15} ) q^{64} + ( 2 \beta_{1} - \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{65} + ( -3 + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{66} + ( -1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{11} + \beta_{15} ) q^{67} + ( 3 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{15} ) q^{68} + ( 2 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{15} ) q^{69} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{70} + ( 2 \beta_{1} - \beta_{9} + 2 \beta_{14} ) q^{71} + ( 6 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{8} ) q^{72} + ( -2 \beta_{1} - 2 \beta_{10} + 2 \beta_{13} ) q^{73} + ( \beta_{1} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{74} + ( -2 \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{75} + ( 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{11} ) q^{76} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{15} ) q^{77} + ( \beta_{1} + 3 \beta_{6} - \beta_{7} + 2 \beta_{9} + 3 \beta_{12} - \beta_{14} ) q^{78} + ( -\beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{14} ) q^{79} + ( -\beta_{1} - \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{80} + ( -1 + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{11} ) q^{81} + ( 2 \beta_{1} + 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{82} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{8} - 2 \beta_{15} ) q^{83} + ( 5 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{11} - \beta_{15} ) q^{84} + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{85} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{11} - \beta_{15} ) q^{86} + ( -5 + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{8} - 2 \beta_{11} ) q^{87} + ( \beta_{1} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{88} + ( -2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{8} + \beta_{11} ) q^{89} + ( -\beta_{1} + 2 \beta_{10} ) q^{90} + ( -2 \beta_{6} + \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{91} + ( -\beta_{1} + \beta_{6} + \beta_{7} + 2 \beta_{10} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{92} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - 4 \beta_{11} - \beta_{15} ) q^{93} + ( -1 + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{8} + 2 \beta_{11} + \beta_{15} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{14} ) q^{95} + ( \beta_{1} - 3 \beta_{6} - \beta_{9} + \beta_{10} - 3 \beta_{12} - \beta_{13} + \beta_{14} ) q^{96} + ( \beta_{7} + 2 \beta_{9} + 3 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{97} + ( 3 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} ) q^{98} + ( 2 \beta_{7} - \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 6 q^{4} - 2 q^{8} + 8 q^{9} + O(q^{10})$$ $$16 q - 2 q^{2} - 6 q^{4} - 2 q^{8} + 8 q^{9} - 8 q^{15} - 14 q^{16} - 2 q^{18} - 16 q^{30} - 2 q^{32} - 8 q^{33} + 18 q^{34} - 22 q^{36} + 36 q^{38} - 24 q^{47} - 8 q^{49} + 34 q^{50} - 8 q^{55} - 16 q^{60} - 30 q^{64} - 32 q^{66} + 38 q^{68} + 40 q^{70} + 70 q^{72} + 4 q^{76} - 24 q^{81} + 72 q^{84} + 4 q^{86} - 40 q^{87} - 24 q^{89} - 16 q^{94} + 34 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 8 x^{14} + 20 x^{12} + 36 x^{10} + 240 x^{8} - 156 x^{6} + 268 x^{4} + 136 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1122577 \nu^{14} - 14068890 \nu^{12} - 66487248 \nu^{10} - 169343412 \nu^{8} - 515961792 \nu^{6} - 1043866460 \nu^{4} + 110695532 \nu^{2} - 581192704$$$$)/ 388027784$$ $$\beta_{3}$$ $$=$$ $$($$$$-1923537 \nu^{14} - 13519726 \nu^{12} - 25288536 \nu^{10} - 46964780 \nu^{8} - 438449040 \nu^{6} + 649223580 \nu^{4} - 1494405076 \nu^{2} + 151276224$$$$)/ 388027784$$ $$\beta_{4}$$ $$=$$ $$($$$$-511505 \nu^{14} - 4434484 \nu^{12} - 12763476 \nu^{10} - 23013159 \nu^{8} - 130749772 \nu^{6} - 14220716 \nu^{4} - 93359810 \nu^{2} - 96545102$$$$)/97006946$$ $$\beta_{5}$$ $$=$$ $$($$$$2530173 \nu^{14} + 18755430 \nu^{12} + 39114176 \nu^{10} + 69125296 \nu^{8} + 597246024 \nu^{6} - 638782644 \nu^{4} + 1098739804 \nu^{2} + 450774640$$$$)/ 388027784$$ $$\beta_{6}$$ $$=$$ $$($$$$-3391627 \nu^{15} - 26976512 \nu^{13} - 68170444 \nu^{11} - 131353936 \nu^{9} - 826357224 \nu^{7} + 587561828 \nu^{5} - 1104732028 \nu^{3} + 49963176 \nu$$$$)/ 388027784$$ $$\beta_{7}$$ $$=$$ $$($$$$-3710839 \nu^{15} - 33502210 \nu^{13} - 107383932 \nu^{11} - 227489312 \nu^{9} - 1047214652 \nu^{7} - 282204156 \nu^{5} - 686733980 \nu^{3} - 14322168 \nu$$$$)/ 388027784$$ $$\beta_{8}$$ $$=$$ $$($$$$4466259 \nu^{14} + 35818322 \nu^{12} + 84156112 \nu^{10} + 113648052 \nu^{8} + 941650648 \nu^{6} - 853219156 \nu^{4} + 85178972 \nu^{2} + 997692448$$$$)/ 388027784$$ $$\beta_{9}$$ $$=$$ $$($$$$-2689779 \nu^{15} - 22018279 \nu^{13} - 58720920 \nu^{11} - 117192984 \nu^{9} - 707674738 \nu^{7} + 203899652 \nu^{5} - 889740780 \nu^{3} - 870945096 \nu$$$$)/ 194013892$$ $$\beta_{10}$$ $$=$$ $$($$$$1397209 \nu^{15} + 12471803 \nu^{13} + 37660840 \nu^{11} + 70747866 \nu^{9} + 364403110 \nu^{7} + 47661826 \nu^{5} - 6379140 \nu^{3} + 588735180 \nu$$$$)/97006946$$ $$\beta_{11}$$ $$=$$ $$($$$$-4414637 \nu^{14} - 35845480 \nu^{12} - 93697396 \nu^{10} - 177380254 \nu^{8} - 1087856768 \nu^{6} + 559120396 \nu^{4} - 1291451648 \nu^{2} - 337140920$$$$)/ 194013892$$ $$\beta_{12}$$ $$=$$ $$($$$$-7976257 \nu^{15} - 58299578 \nu^{13} - 116440672 \nu^{11} - 185835896 \nu^{9} - 1741523828 \nu^{7} + 2539117108 \nu^{5} - 3208985236 \nu^{3} + 205818936 \nu$$$$)/ 388027784$$ $$\beta_{13}$$ $$=$$ $$($$$$-9891357 \nu^{15} - 76989604 \nu^{13} - 183627060 \nu^{11} - 339496648 \nu^{9} - 2385051376 \nu^{7} + 1818685188 \nu^{5} - 4071316148 \nu^{3} - 1023743432 \nu$$$$)/ 388027784$$ $$\beta_{14}$$ $$=$$ $$($$$$10838191 \nu^{15} + 87123872 \nu^{13} + 214521736 \nu^{11} + 348654544 \nu^{9} + 2465582864 \nu^{7} - 1873346700 \nu^{5} + 1381761084 \nu^{3} + 1007435080 \nu$$$$)/ 388027784$$ $$\beta_{15}$$ $$=$$ $$($$$$-12834559 \nu^{14} - 99013618 \nu^{12} - 221895280 \nu^{10} - 344990604 \nu^{8} - 2849496608 \nu^{6} + 3063239676 \nu^{4} - 2688584780 \nu^{2} - 1418449856$$$$)/ 388027784$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{8} - \beta_{5} - 3 \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{14} - \beta_{13} - 2 \beta_{12} + 3 \beta_{10} + 2 \beta_{9} + 3 \beta_{6} - 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{15} - 3 \beta_{11} + 4 \beta_{5} - \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{14} - 2 \beta_{13} + 9 \beta_{12} - 7 \beta_{10} - 2 \beta_{9} + 3 \beta_{7} - 11 \beta_{6} + 6 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{15} + 16 \beta_{11} - 12 \beta_{8} - 13 \beta_{5} - 13 \beta_{4} - 12 \beta_{3} - 10 \beta_{2} + 3$$ $$\nu^{7}$$ $$=$$ $$-12 \beta_{14} + 13 \beta_{13} - 38 \beta_{12} + 13 \beta_{10} - 5 \beta_{9} - 27 \beta_{7} + 72 \beta_{6} - 13 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$68 \beta_{15} - 53 \beta_{11} + 95 \beta_{8} + 109 \beta_{5} + 114 \beta_{4} + 27 \beta_{3} + 9 \beta_{2} - 98$$ $$\nu^{9}$$ $$=$$ $$\beta_{14} - 42 \beta_{13} + 137 \beta_{12} + 26 \beta_{10} + 18 \beta_{9} + 189 \beta_{7} - 389 \beta_{6} - 103 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-337 \beta_{15} + 60 \beta_{11} - 671 \beta_{8} - 627 \beta_{5} - 627 \beta_{4} + 171 \beta_{3} + 148 \beta_{2} + 857$$ $$\nu^{11}$$ $$=$$ $$276 \beta_{14} + 232 \beta_{13} - 398 \beta_{12} - 610 \beta_{10} - 202 \beta_{9} - 1120 \beta_{7} + 1682 \beta_{6} + 1322 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$1346 \beta_{15} + 812 \beta_{11} + 4020 \beta_{8} + 2292 \beta_{5} + 3408 \beta_{4} - 2900 \beta_{3} - 1858 \beta_{2} - 4928$$ $$\nu^{13}$$ $$=$$ $$-3152 \beta_{14} - 1094 \beta_{13} - 460 \beta_{12} + 5826 \beta_{10} + 2058 \beta_{9} + 5418 \beta_{7} - 5392 \beta_{6} - 9842 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-3052 \beta_{15} - 10250 \beta_{11} - 19406 \beta_{8} - 5826 \beta_{5} - 15868 \beta_{4} + 22862 \beta_{3} + 14574 \beta_{2} + 24820$$ $$\nu^{15}$$ $$=$$ $$23600 \beta_{14} + 2314 \beta_{13} + 17496 \beta_{12} - 39954 \beta_{10} - 12460 \beta_{9} - 20804 \beta_{7} + 4094 \beta_{6} + 60402 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.970055 + 0.510012i −0.970055 − 0.510012i 0.970055 − 0.510012i −0.970055 + 0.510012i −0.289081 − 0.578468i 0.289081 + 0.578468i −0.289081 + 0.578468i 0.289081 − 0.578468i −0.476789 + 2.28924i 0.476789 − 2.28924i −0.476789 − 2.28924i 0.476789 + 2.28924i 1.32216 − 1.07919i −1.32216 + 1.07919i 1.32216 + 1.07919i −1.32216 − 1.07919i
−1.25175 0.658116i −0.909242 1.13377 + 1.64760i −1.54992 1.13815 + 0.598386i 3.57366i −0.334887 2.80853i −2.17328 1.94011 + 1.02002i
101.2 −1.25175 0.658116i 0.909242 1.13377 + 1.64760i 1.54992 −1.13815 0.598386i 3.57366i −0.334887 2.80853i −2.17328 −1.94011 1.02002i
101.3 −1.25175 + 0.658116i −0.909242 1.13377 1.64760i −1.54992 1.13815 0.598386i 3.57366i −0.334887 + 2.80853i −2.17328 1.94011 1.02002i
101.4 −1.25175 + 0.658116i 0.909242 1.13377 1.64760i 1.54992 −1.13815 + 0.598386i 3.57366i −0.334887 + 2.80853i −2.17328 −1.94011 + 1.02002i
101.5 −0.632188 1.26504i −2.88107 −1.20068 + 1.59949i 0.914541 1.82138 + 3.64469i 2.61974i 2.78248 + 0.507728i 5.30059 −0.578162 1.15694i
101.6 −0.632188 1.26504i 2.88107 −1.20068 + 1.59949i −0.914541 −1.82138 3.64469i 2.61974i 2.78248 + 0.507728i 5.30059 0.578162 + 1.15694i
101.7 −0.632188 + 1.26504i −2.88107 −1.20068 1.59949i 0.914541 1.82138 3.64469i 2.61974i 2.78248 0.507728i 5.30059 −0.578162 + 1.15694i
101.8 −0.632188 + 1.26504i 2.88107 −1.20068 1.59949i −0.914541 −1.82138 + 3.64469i 2.61974i 2.78248 0.507728i 5.30059 0.578162 1.15694i
101.9 0.288356 1.38450i −1.14379 −1.83370 0.798461i −3.30694 −0.329818 + 1.58358i 1.93801i −1.63423 + 2.30853i −1.69175 −0.953577 + 4.57847i
101.10 0.288356 1.38450i 1.14379 −1.83370 0.798461i 3.30694 0.329818 1.58358i 1.93801i −1.63423 + 2.30853i −1.69175 0.953577 4.57847i
101.11 0.288356 + 1.38450i −1.14379 −1.83370 + 0.798461i −3.30694 −0.329818 1.58358i 1.93801i −1.63423 2.30853i −1.69175 −0.953577 4.57847i
101.12 0.288356 + 1.38450i 1.14379 −1.83370 + 0.798461i 3.30694 0.329818 + 1.58358i 1.93801i −1.63423 2.30853i −1.69175 0.953577 + 4.57847i
101.13 1.09558 0.894257i −1.88797 0.400610 1.95947i 2.41361 −2.06843 + 1.68833i 2.57100i −1.31336 2.50501i 0.564439 2.64431 2.15839i
101.14 1.09558 0.894257i 1.88797 0.400610 1.95947i −2.41361 2.06843 1.68833i 2.57100i −1.31336 2.50501i 0.564439 −2.64431 + 2.15839i
101.15 1.09558 + 0.894257i −1.88797 0.400610 + 1.95947i 2.41361 −2.06843 1.68833i 2.57100i −1.31336 + 2.50501i 0.564439 2.64431 + 2.15839i
101.16 1.09558 + 0.894257i 1.88797 0.400610 + 1.95947i −2.41361 2.06843 + 1.68833i 2.57100i −1.31336 + 2.50501i 0.564439 −2.64431 2.15839i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.b even 2 1 inner
136.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.h.a 16
3.b odd 2 1 1224.2.l.b 16
4.b odd 2 1 544.2.h.a 16
8.b even 2 1 inner 136.2.h.a 16
8.d odd 2 1 544.2.h.a 16
12.b even 2 1 4896.2.l.b 16
17.b even 2 1 inner 136.2.h.a 16
24.f even 2 1 4896.2.l.b 16
24.h odd 2 1 1224.2.l.b 16
51.c odd 2 1 1224.2.l.b 16
68.d odd 2 1 544.2.h.a 16
136.e odd 2 1 544.2.h.a 16
136.h even 2 1 inner 136.2.h.a 16
204.h even 2 1 4896.2.l.b 16
408.b odd 2 1 1224.2.l.b 16
408.h even 2 1 4896.2.l.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.h.a 16 1.a even 1 1 trivial
136.2.h.a 16 8.b even 2 1 inner
136.2.h.a 16 17.b even 2 1 inner
136.2.h.a 16 136.h even 2 1 inner
544.2.h.a 16 4.b odd 2 1
544.2.h.a 16 8.d odd 2 1
544.2.h.a 16 68.d odd 2 1
544.2.h.a 16 136.e odd 2 1
1224.2.l.b 16 3.b odd 2 1
1224.2.l.b 16 24.h odd 2 1
1224.2.l.b 16 51.c odd 2 1
1224.2.l.b 16 408.b odd 2 1
4896.2.l.b 16 12.b even 2 1
4896.2.l.b 16 24.f even 2 1
4896.2.l.b 16 204.h even 2 1
4896.2.l.b 16 408.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + 8 T + 8 T^{2} + 4 T^{3} + 4 T^{4} + 2 T^{5} + 2 T^{6} + T^{7} + T^{8} )^{2}$$
$3$ $$( 32 - 76 T^{2} + 56 T^{4} - 14 T^{6} + T^{8} )^{2}$$
$5$ $$( 128 - 240 T^{2} + 120 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$7$ $$( 2176 + 1396 T^{2} + 316 T^{4} + 30 T^{6} + T^{8} )^{2}$$
$11$ $$( 32 - 268 T^{2} + 408 T^{4} - 42 T^{6} + T^{8} )^{2}$$
$13$ $$( 4352 + 5872 T^{2} + 992 T^{4} + 56 T^{6} + T^{8} )^{2}$$
$17$ $$( 83521 + 3468 T^{2} - 1632 T^{3} + 102 T^{4} - 96 T^{5} + 12 T^{6} + T^{8} )^{2}$$
$19$ $$( 4352 + 7168 T^{2} + 1696 T^{4} + 80 T^{6} + T^{8} )^{2}$$
$23$ $$( 8704 + 19508 T^{2} + 2892 T^{4} + 106 T^{6} + T^{8} )^{2}$$
$29$ $$( 445568 - 89840 T^{2} + 5592 T^{4} - 132 T^{6} + T^{8} )^{2}$$
$31$ $$( 8704 + 77652 T^{2} + 8284 T^{4} + 178 T^{6} + T^{8} )^{2}$$
$37$ $$( 682112 - 143344 T^{2} + 8392 T^{4} - 164 T^{6} + T^{8} )^{2}$$
$41$ $$( 8912896 + 769280 T^{2} + 22528 T^{4} + 264 T^{6} + T^{8} )^{2}$$
$43$ $$( 17408 + 18944 T^{2} + 3776 T^{4} + 124 T^{6} + T^{8} )^{2}$$
$47$ $$( -256 - 216 T - 28 T^{2} + 6 T^{3} + T^{4} )^{4}$$
$53$ $$( 69632 + 157440 T^{2} + 11136 T^{4} + 208 T^{6} + T^{8} )^{2}$$
$59$ $$( 17408 + 27904 T^{2} + 5120 T^{4} + 188 T^{6} + T^{8} )^{2}$$
$61$ $$( 15488 - 62064 T^{2} + 7416 T^{4} - 164 T^{6} + T^{8} )^{2}$$
$67$ $$( 526592 + 350976 T^{2} + 15648 T^{4} + 224 T^{6} + T^{8} )^{2}$$
$71$ $$( 34816 + 135860 T^{2} + 9852 T^{4} + 182 T^{6} + T^{8} )^{2}$$
$73$ $$( 67403776 + 4326400 T^{2} + 73088 T^{4} + 464 T^{6} + T^{8} )^{2}$$
$79$ $$( 263296 + 155252 T^{2} + 14908 T^{4} + 266 T^{6} + T^{8} )^{2}$$
$83$ $$( 5030912 + 551936 T^{2} + 19840 T^{4} + 268 T^{6} + T^{8} )^{2}$$
$89$ $$( 8 + 164 T - 96 T^{2} + 6 T^{3} + T^{4} )^{4}$$
$97$ $$( 117121024 + 4580608 T^{2} + 66496 T^{4} + 424 T^{6} + T^{8} )^{2}$$